Positive codegree thresholds for Hamilton cycles in hypergraphs

For each $k\ge 3$ and $1\le \mathrm{\ell }\le k-1$ we give an asymptotically best possible minimum positive codegree condition for the existence of a Hamilton $\mathrm{\ell }$-cycle in a $k$-uniform hypergraph. This result exhibits an interesting duality with its analogue under a minimum codegree condition. The special case $\mathrm{\ell }=k-1$ of our result establishes an asymptotic version of a recent conjecture of Illingworth, Lang, Müyesser, Parczyk and Sgueglia on tight Hamilton cycles in hypergraphs.